Catenary ring: Difference between revisions

Latest revision as of 16:02, 18 July 2013

This article defines a property of commutative unital rings; a property that can be evaluated for a commutative unital ring
View all properties of commutative unital rings
VIEW RELATED: Commutative unital ring property implications | Commutative unital ring property non-implications |Commutative unital ring metaproperty satisfactions | Commutative unital ring metaproperty dissatisfactions | Commutative unital ring property satisfactions | Commutative unital ring property dissatisfactions

Definition

A commutative unital ring is termed a catenary ring or chain ring or is said to satisfy the Saturated Chain Condition if it is Noetherian and satisfies the following equivalent conditions:

If $P<P_{1}<P_{2}<Q$ is a strictly ascending chain of prime ideals, and $P'$ is a prime ideal between $P$ and $Q$ , then there is either a prime ideal between $P$ and $P'$ or a prime ideal between $P'$ and $Q$
Given two prime ideals $P$ and $Q$ such that $P\subset Q$ , the length of any saturated chain of primes between $P$ and $Q$ (i.e. a chain of primes in which no more primes can be inserted in between) is determined independent of the choice of chain

Note that being catenary does not guarantee that any two fully saturated chains of primes (i.e. any two chains of primes which are saturated and cannot be extended in either direction) have the same length. The problem is that the starting and ending points of the chains may differ: there may be many different maximal ideals and many different minimal prime ideals.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
polynomial ring over a field	of the form $K[x]$ where $K$ is a field			click here
multivariate polynomial ring over a field	of the form $K[x_{1},x_{2},\dots ,x_{n}]$ where $K$ is a field			click here
affine ring over a field		affine implies catenary		click here
principal ideal domain	integral domain in which every ideal is a [principal ideal]]		click here
universally catenary ring	every finitely generated algebra over it is a catenary ring.			click here
Cohen-Macaulay ring	for every ideal, the depth equals the codimension			click here

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Noetherian ring	every ideal in it is finitely generated			click here

Metaproperties

Closure under taking quotient rings

This property of commutative unital rings is quotient-closed: the quotient ring of any ring with this property, by any ideal in it, also has this property

View other quotient-closed properties of commutative unital rings

@@ Line 1: / Line 1: @@
-{{commring property}}
+{{curing property}}
 ==Definition==
-A [[commutative unital ring]] is said to be '''catenary''' if it is [[Noetherian ring|Noetherian]] satisfies the following condition:
+A [[commutative unital ring]] is termed a '''catenary ring''' or '''chain ring''' or is said to satisfy the '''Saturated Chain Condition''' if it is [[Noetherian ring|Noetherian]] and satisfies the following equivalent conditions:
-If <math>P < P_1 < P_2 < Q</math> is a strictly ascending chain of [[prime ideal]]s, and <math>P'</math> is a prime ideal between <math>P</math> and <math>Q</math>, then there is either a prime ideal between <math>P</math> and <math>P'</math> or a prime ideal between <math>P'</math> and <math>Q</math>.
+* If <math>P < P_1 < P_2 < Q</math> is a strictly ascending chain of [[prime ideal]]s, and <math>P'</math> is a prime ideal between <math>P</math> and <math>Q</math>, then there is either a prime ideal between <math>P</math> and <math>P'</math> or a prime ideal between <math>P'</math> and <math>Q</math>
+* Given two prime ideals <math>P</math> and <math>Q</math> such that <math>P \subset Q</math>, the length of any saturated chain of primes between <math>P</math> and <math>Q</math> (i.e. a chain of primes in which no more primes can be inserted in between) is determined independent of the choice of chain
+Note that being catenary does ''not'' guarantee that any two fully saturated chains of primes (i.e. any two chains of primes which are saturated and cannot be extended in either direction) have the same length. The problem is that the starting and ending points of the chains may differ: there may be many different [[maximal ideal]]s and many different [[minimal prime ideal]]s.
 ==Relation with other properties==
@@ Line 11: / Line 14: @@
 ===Stronger properties===
-* [[Affine ring]]
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::polynomial ring over a field]] || of the form <math>K[x]</math> where <math>K</math> is a [[field]] || || || {{intermediate notions short|catenary ring|polynomial ring over a field}}
+|-
+| [[Weaker than::multivariate polynomial ring over a field]] || of the form <math>K[x_1,x_2,\dots,x_n]</matH> where <math>K</math> is a [[field]] || || || {{intermediate notions short|catenary ring|multivariate polynomial ring over a field}}
+|-
+| [[Weaker than::affine ring over a field]] || || [[affine implies catenary]] || || {{intermediate notions short|catenary ring|affine ring over a field}}
+|-
+| [[Weaker than::principal ideal domain]] || [[integral domain]] in which every [[ideal]] is a [principal ideal]] || || {{intermediate notions short|catenary ring|principal ideal domain}}
+|-
+| [[Weaker than::universally catenary ring]] || every finitely generated algebra over it is a catenary ring.|| || || {{intermediate notions short|catenary ring|universally catenary ring}}
+|-
+| [[Weaker than::Cohen-Macaulay ring]] || for every ideal, the depth equals the codimension || || || {{intermediate notions short|catenary ring|Cohen-Macaulay ring}}
+|}
+===Weaker properties===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::Noetherian ring]] || every [[ideal]] in it is [[finitely generated ideal|finitely generated]] || || || {{intermediate notions short|Noetherian ring|catenary ring}}
+|}
+==Metaproperties==
+{{Q-closed curing property}}